A 2007 plane-wave density-functional-theory study of free-standing Be(0001) slabs of up to 25 monolayers found that quantum size effects make the surface energy, work function, interlayer relaxation, and atomic-hydrogen binding energy oscillate with film thickness; that hydrogen adsorption shifts the surface electronic structure upward toward a narrow density-of-states peak until the shift saturates near 10 monolayers; and that the thickness dependence of the H binding energy closely resembles that of the work function, a similarity whose cause the authors explicitly stated remained unclear. That study, restricted to ideal, defect-free, free-standing slabs at zero temperature computed with a single specific computational setup, left open the microscopic cause of this similarity and what the hydrogen-induced change of the surface density of states implies for the anomalously enhanced electron-phonon coupling and superconducting tendencies of beryllium surfaces. The problem is to establish, quantitatively and mechanistically, the quantum-size-effect-controlled adsorption of hydrogen on Be(0001) films: the mechanism behind the work-function/binding-energy similarity, together with the consequences of the hydrogen-induced density-of-states change for surface electronic properties.

Background

When a metal film is only a few to a few tens of atomic layers thick, electrons confined between its two surfaces form standing waves, called quantum well states, whose allowed energies shift each time the film thickness changes by one monolayer (ML). The resulting quantum size effect (QSE) makes surface energy, work function (the minimum energy needed to remove an electron from the surface), interlayer spacings, and adsorption energies oscillate with film thickness; it was analyzed in early slab calculations for simple metals and observed prominently in Pb films on Si(111) and Cu(111). Beryllium is an unusual simple metal in this context. Bulk Be has a deep minimum in its electronic density of states (DOS) at the Fermi level EFE_F and is nearly semiconducting, whereas the DOS at EFE_F of the Be(0001) surface is larger by a factor of 4-5, making that surface a two-dimensional free-electron-like metal; the surface also shows an anomalous outward expansion of its first interlayer spacing, opposite to the contraction on most metal surfaces. Photoemission experiments and first-principles calculations further showed that the electron-phonon coupling, the interaction between conduction electrons and lattice vibrations that controls superconductivity in conventional superconductors, is strongly enhanced at the Be(0001) surface; this enhancement has been invoked to explain the much higher superconducting transition temperature of amorphous Be compared with crystalline Be reported in the surface-science literature. These facts make Be(0001) films an extreme case for confinement physics: quantum confinement in a slab acts on a surface that is already strongly metallic while the bulk is not. In 2007, Ping Zhang and Hong-Zhou Song reported plane-wave density-functional-theory (DFT) calculations, using ultrasoft pseudopotentials and the PW91 generalized-gradient approximation (GGA) with a 300 eV cutoff, an 11×11×111\times11\times1 k-point mesh, and 20 Å of vacuum, for free-standing Be(0001) slabs of 2 to 25 ML. They found damped oscillations with thickness in the surface energy, the work function, the interlayer relaxation, and the binding energy of an atomic hydrogen monolayer (about 2.2-2.3 eV, with a clear preference for the bridge adsorption site over fcc, hcp, and on-top sites). Hydrogen adsorption shifts the Fermi level upward toward a narrow peak of the clean-surface DOS, and this shift saturates once the film reaches about 10 ML. They also noted that the thickness dependence of the H binding energy closely tracks that of the work function, while stating explicitly that the cause of this similarity remained unclear. The study was restricted to ideal, defect-free, free-standing slabs at zero temperature, computed with a single specific computational setup, and it suggested but did not compute the consequences of the hydrogen-induced DOS change for surface electron-phonon coupling and superconducting tendencies. Hydrogen uptake by beryllium surfaces is also a long-standing concern for plasma-facing components in fusion devices, where beryllium has served as a first-wall material, so the quantitative energetics of H on Be surfaces has applied as well as fundamental interest.

Problem Statement

Establish the quantum-size-effect physics of atomic-hydrogen adsorption on Be(0001) thin films, with one common objective: a quantitative and mechanistic account of the oscillatory thickness dependence of hydrogen adsorption on Be(0001) films, together with its consequences for surface electronic properties. The required results are: (1) identify the microscopic mechanism responsible for the similarity between the thickness dependence of the work function of the clean slab and that of the hydrogen binding energy, establishing by a quantitative comparison that the two share a common oscillation period and phase, which the source explicitly left unexplained, for the oscillatory thickness dependence reported for free-standing Be(0001) slabs of up to 25 monolayers with an adsorbed atomic-hydrogen monolayer; and (2) compute the consequences of the hydrogen-induced modification of the Be(0001) surface density of states, including the reported saturation of the hydrogen-induced electronic-structure changes near N∼10N\sim10 monolayers, for the surface electron-phonon coupling strength and the superconducting transition temperature, which the source speculated about but did not calculate. Because the source results were obtained for ideal, defect-free, free-standing slabs at zero temperature with a single exchange-correlation treatment, how a supporting substrate, finite temperature, realistic growth conditions, surface defects, methodological choices, and beyond-DFT corrections alter the oscillatory trends is itself unresolved; this is retained as background context for such an account, not as an additional required result.

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